Metamaterials offer an enormous degree of freedom for manipulating electromagnetic fields, as independent and nearly arbitrary gradients can be introduced in the components of the effective permittivity and permeability tensors. In order to exploit such a high degree of freedom, a viable method for the well-aimed design of complex materials would be desirable.
Pendry et al., Science 312, 1780 (2006) reported a methodology based on continuous form-invariant coordinate transformations of Maxwell's equations which allows for the manipulation of electromagnetic fields in a previously unknown and unconventional fashion. This method was successfully applied for the design and the experimental realization of an invisibility cloak and generated widespread interest specifically in the prospects of electromagnetic cloaking—a topic that has dominated much of the subsequent discussion.
The methodology presented in Pendry et al makes use of form-invariant continuous coordinate transformations of Maxwell's equations. The use of continuous transformations provides a complex transformation-optical material which is invisible to an external observer. In other words, the field modifications precipitated in the transformation-optical device generally may not be transferred to another medium and the original electromagnetic properties of waves impinging on the medium are restored as soon as the waves exit the optical component. Transformation-optical designs reported in the literature so far generally have in common that the electromagnetic properties of the incident waves are exclusively changed within the restricted region of the transformation-optical device. However, for the sake of the continuity of the transformation, the field manipulation cannot be transferred to another medium or free space and thus remains an, in many cases, undesired local phenomenon.
It would be desirable to have also a tool for the design of electromagnetic/optical components that takes advantage of the high degree of design freedom provided by the transformation-optical approach, but allows the transfer of field manipulations outside the transformation-optical material. Such a method would allow the creation of optical devices with unconventional electromagnetic/optical behavior and functionality that exceeds the abilities of conventional components like lenses, beam steerers, beam shifters, beam splitters and similar.
The technology herein expands the design method for complex electromagnetic materials from form-invariant coordinate transformations of Maxwell's equations to finite embedded coordinate transformations. In contrast to continuous transformations, embedded transformations allow the transfer of electromagnetic field manipulations from the transformation-optical medium to another medium, thereby allowing the design of structures that are not exclusively invisible. The illustrative exemplary non-limiting implementations provide methods to design such novel devices by a modified transformation-optical approach. The conceived electromagnetic/optical devices can be reflectionless under certain circumstances.
The exemplary illustrative non-limiting technology herein further delivers a topological criterion for the reflectionless design of complex media. This exemplary illustrative non-limiting expanded method can be illustrated in conjunction with the topological criterion to provide an example illustrative non-limiting parallel beam shifter and beam splitter with unconventional electromagnetic behavior.
The concept of embedded coordinate transformations significantly expands the idea of the transformation-optical design of metamaterials which itself was restricted to continuous coordinate transformations so far. The expansion to embedded transformations allows for non-reversibly change to the properties of electromagnetic waves in transformation media and for transmission of the changed electromagnetic properties to free space or to a different medium in general. In order to design the medium as reflectionless, a new topological criterion for the embedded transformations can be used to impose constraints to the metric of the spaces at the interface between the transformation-optical medium and the surrounding space. This metric criterion can be applied in the conception of a parallel beam shifter and a beam splitter and confirmed in 2D full wave simulations. Such exemplary illustrative non-limiting devices can provide an extraordinary electromagnetic behavior which is not achievable by conventional materials. Such examples clearly state the significance of embedded coordinate transformations for the design of new electromagnetic elements with tunable, unconventional optical properties.